AN EXAMPLE: DISCOVERY OF UNIQUE FACTORIZATION

Suppose that the  system discovers the  concept of PRIMES, much  as I
outlined near  the beginning of this talk. As  you may recall, one of
AM's heuristics that  adds heuristics then said  the following was  a
new rule of thumb to be followed henceforth:


⊗2"Primes are  useful to generate  and to test  conjectures involving
multiplication and division"⊗*

We'll use that heuristic in a minute.

Say that AM is trying to judge how interesting the concept FACTORS-OF
is.   AM  assembles  heuristics  from  the Interest  facets  of  that
concept,     and     all     its    generalizations:     (FACTORS-OF,
INVERTED-OPERATION, OPERATION, ACTIVITY, ANY-CONCEPT, ANYTHING).

In particular, OPERATION.INT  contains a rule  that any operation  is
interesting if the images of the domain elements, taken together as a
set, are interesting.

We  are  trying  to see  why  the  set of  all  the  images might  be
interesting.  Well, why is a set interesting?  AM gathers up  all the
heuristics         from        (SET.INT,         UNORDERED-STRUC.INT,
NONMULT-ELES-STRUC.INT,  STRUCTURE.INT,  OBJECT.INT, ANY-CONCEPT.INT,
ANYTHING.INT).

A structure is interesting  if each element is mildly  interesting in
precisely the same way.

So AM  looks at the examples  it has of FACTORS-OF,  <slide> at their
image sets, to tries to notice some regularity.

To see if  all the members  of a structure  satisfy some  interesting
property, one strategy is to list all the interesting properties that
oneof them satisfies, then check those against the other elements.

So, recursing,  AM tries to find properties  that one of these images
satisfies, say this one {(BAG 17 1)}.

If the system has the  concept SINGLETON, then this is proposed,  but
it fails to be satisfied by the other image sets.

From  here  again  (STRUC.INT),  AM   finds  that  a  set  is  mildly
interesting if one element is very interesting.

So how is (BAG 17 1) interesting?

Say the  system has the concept doubleton. Then this is proposed, and
succeeds, and AM conjectures that all numbers can be  factored as the
product of  two numbers. A new relation  is created, which associates
to each  number, all  the  possible ways  of  factoring it  into  the
product of two numbers.

AM  then continues  on,  since its  time-slice  isn't exhausted  yet,
looking for more ways in which this bag could be interesting.

From here again,  a structure is interesting if each element has some
rare property in common.  Well, all these elements are odd,  and also
they're all primes, and one of them is the number itself.

Here's a counterexample to factoring  into odd numbers, but the other
two  observations seem generally applicable.   So AM conjectures that
every number can be factored as 1 times some  other numbers, which is
not  very profound,  and  also conjectures  that each  number  can be
factored into he product of a bunch of primes.

A new relation  R is created,  which associates to  each number,  all
possible  ways  of  factoring that  number  into  primes.  The  first
question AM  asks about R is whether or not  is a function.  That is,
is it defined everywhere and  is it single-valued.  This simple  idea
-- That relation R is a function  -- is the complete statement of the
unique factorization theorem.

END OF UFT EXAMPLE

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